Corresponding states relations of linear dimethysiloxane oligomers. I. Pure components

Chemical Physics 10 (1975) 17-22 Q North-Holland Publishing Company

CORRESPONDING STATES RELATIONS OF LINEAR DIMETHYUILOXANE

OUGOMERS.

1. PURE COMPONENTS Eric DICKINSON, Ian A. McLURE, Alistair J. PRETTY and Paul A. SADLER Depnnment of (;hemisrry, i%e Univem’ty. Sheffield. S3 7HF, UK Received 14 January 1975

Corresponding states relationships are tested for the linear dimethylsiloxane oligomers containing from two to six silicon atoms. From an analysis of liquid state p, V, Tdata over the temperature range 278-358 K for the dimer and 298-408 K for the other siloxanes. relative values of temperature, pressure. volume and entropy reduction parameters a12 derived. The Set of linear dimerhylsiloxanes is found 10 be well represented by a principle of corresponding states, but it is one which is slightly, albeit significantly, different from that which describes the behaviour of the normsl alkaner

1. Introduction The principle of corresponding states is a central assumption of many theories of chain molecule liquid mixtures. It has been tested in detail, however, for only one set of linear homologues - the normal alkanes. The linear dimethylsiloxanes, (CH,),SiO{(CH,),SiO),_2Si(CH3)3 (n 2 0), have received little attention [l], although in principle they can provide a more exacting test of corresponding states, on account of their greater liquid range. We have been interested recently in the thermodynamic behaviour of binary liquid mixtures of n-alkanes with linear dimethylsiloxanes [2,3], Since an interpretation of these mixtures in terms of current statistical thermodynamic approaches [4-61 invariably involves assuming conformality of the intermolecular pair potential functions, in order to assess properly any conclusions derived from such a study, it is necessary to know how far the principle of corresponding states is applicable to the systems under investigation. We must test the principle, therefore, firstly as applied to the set of pure linear dimethylsiloxane oligomers, and, secondly, as applied to that larger set comprising both linear siloxanes and normal &canes. Patterson and Bardin [7] have tested the corresponding states principle for the n-alkanes from methane to polymethylene. Their treatment is essentially pheno-

menological, requiring no fluid model or empirical equation of state. The characteristic parameters are determined from Prigogine’s formulation for the corresponding states relations of a series of chain molecule liquids [6]. To facilitate comparison with the Ralkane behaviour, we employ a similar procedure in this paper to analyse our measurements of dimethylsiloxane molar volumes V, [81, isobaric expansivities [8] and thermal pressure coefficients 7~ [9]. The 2 ermodynamic reduction parameters thereby derivecr

are important ties [2].

in the prediction

2. Temperature

of the mixing proper-

reduction factors

The reduced expansivity of an oligomeric species of chain length n is given by the relation S(B=o&T)TYn).

0)

where 7 = T/T* is the reduced temperature and T* is a temperature reduction factor which depends only upon the chain length. The dimensionless quantity opT_is already in reduced form, and (being a function of T) is therefore a useful measure of the effective reduced temperature. A plot of (%7’)” against log10(7’/K) for each siloxane gives a series of curves (fig. 1) which may be regarded as straight lines over

E. Dickinson et d.ISUoxane corresponding states

rcii~~ns

2.7

1.7 -

f

1.2. 244

249

2.59

2.54

2.64

log,JT/K) Fig. 1. Plots of $,T)-’

against logto(T/K):

(a) dimer.

(b) trimer. Cc) tetramer,

the temperature range of the data. The coefficients A and B in the equation @Pn

-l = A - Bloglo(T/K)

(2)

are listed in table 1. The relative temperature reduction factor for a given siloxane (S, n) is obtained by translating eq. (2) through the distance logto(T&JT&), where Tzf is the temperature reduction factor for an appropriate conformal reference substance. If the linear dimethylsiloxanes were to obey corresponding states exactly, the slopes of the curves in fig. 1 would be identical. Although B does decrease slightly with increasing n (a similar trend occurs also with the n-alkanes [7]), table 1 shows that this requirement is reasonably well fulfilled. Any on; of the siloxanes could act equally well as arbitrary reference substance, but we choose n-octane (A,@ so as to comply

(d) pentamer, (e) hexamer. (f) n-octane.

with the previous n-alkane analysis [7]. Unfortunately, as the n-octane line of fig. 1 is by no means parallel to the siloxane curves, we are forced to conclude that the hydrocarbon follows a slightly different corresponding states principle from that of the siloxanes. A corollary of this is the fact that the ratio Tsf,,,/T& is not independent of temperature, as it should be under ideally conformal conditions. The temperature reduction factors recorded in table 2 were evaluated a value of (“pT>-’ at which there at(cx 7)-l =20P . is good overlap of the different siloxane reduced temperatures t . f The values of Tg,n/Ti,g

listed in table 2 differ slightly from those given in ref. [2], which were evaluated at $J’J-’ = 3.0 from p, K Tdata available over a much shorter temperature range (ref. [lo]). In terms of the ratios Ti ,,/T< 2, however, the two sets of reduction factors agree to hhd

t I per cent.

Tab!e 1 The coefficients of eqs. (2). (3) and (6) for the linear dimethylsiloxanes Oligomex

A

E

c

D

E

F

dlnter

tier tebamu

25.64 25.13 24.15

9.39 9.07 8.85

5.548 5.558 5.587

0.584 0.652

2.2354 2.3855

0.2238 0.2133

pentamer

24.49

8.69

5.609

hcxamer

24.30

8.57

5.631

0.748 0.835 0.908

2.4806 2.5565 2.6256

0.2441 0.2722 0.2813

E. Dickinson et alhloxane Table 2 The reduction factors of the linear dimethylsiloxanes relawe to those of n-octane (A.8)

~g,~/pi,g listed in table 2 were evaluated at %T= 0.42 by shifting the curves through a distance

(S,n)

~ofzlo(Ps’,“/P~.lJ>

Oligomer

Ts’. fP~,s

&,nlP~,S

G’,n%,8

diner tsimer tetramer pentamer

0.903 0.969 1.022 1.062

0.836 0.800 0.780 0.755

1.252 1.141 2.254 2.772

hexamer

1.094

0.139

3.285

Using isothermal compressibilities calculated from experimental values of c”p and 7v, Patterson and Bardin [7] determined pressure reduction factors for the n-tines from plots of loglo@T/kN-l m2) against QPT. ln our analysis of the dimethylsiloxane data, we prefer to use the thermal pressure coefficients directly. Fig. 2 shows plots of logtO(TrV/kN mS2) against %T for each siloxane. and table 1 lists the coefficients C and D in the equation me2) = C-

D(%T,.

p*(n) = q(n) c/r(n) u3 .

(4)

E and u are characteristic energy and size parameters, and q(n) and r(n) are effective numbers of segments proportional respectively to the molecular su$ace area and molecular volume. Without having to define explicitly what is meant by a “segment” of a chain molecule, it is reasonable to assume that such an entity in the middle of the molecule is characterized by an intrinsically lower value of q than one at the end.

(3)

The values of the relative pressure reduc:ion

i.

Three features stand out from the curves of fig. 2: (i) the decrease in the siloxane pressure reduction factor with increasing chain length, (ii) the significantly lower pressure reduction factors for the siloxanes as compared with the n-alkanes (pz n is essentially independent of n [7]), and (iii) the’increasing differences between the pressure reduction factors of the inditidual siloxanes as the reduced temperature is raised. The decrease of p* with increasing chain length can be explained in terms of Ptigogine’s expression [6] :

3. Pressure reduction factors

log~o(T~&N

19

corresponding states rehbns

t The parameters of ref. [2] were evaluated slightly different-

factors

ly (see ref. [lo])

from those reported here.

b, 0.4

0.3

0.5

0.6

0.7

o&T Fig. 2. plots of logm(Trv/LN

rn?

against ccpT: (a) dim% (b) trimer,(c) tctramcr. (d) pentamer, (e) hexamer, (f) aacme.

20

6. Dickihon et ai./Siloxane cotresponding states &tions

.That is, other things being equal, a lower value of p*

is to be expected for molecules of greater chain length. Whilst this reduction in p*(n) with increasing n is observed here for the dimethylsiloxanes, it appears not to be fqund with the n-alkanes [7], where pf; n is almost independent of n. One plausible expla&ion for the n-alkane case concerns a possible offsetting of the larger number of contacts at the n-alkane end segment by a lower intermolecular energy Ed. Alternatively, however, if the reduction parameters are evaluated from Fiery’s reduced equation of state [ 111 7/p=

‘y’l/3/(jTl/3_])

_ (W-1

,

(9

whereas p& is found to be almost independent of chain length [ 121, &, on the other hand, increases significantly with n [I 11. What is clear from the above observations is that the importance, or otherwise, of so-called endeffects in chain molecule liquid mixtures (as indicated by variations in the parameter p*) is critically dependent upon the form of the chosen equation of state. n 4. Volume reduction factors Aplot of log~o(Y,,,/cm3mol-*) against %Tfor each siloxane is almost linear over the temperature range considered. Values of the coefficients E and F in the relation F(%?‘J

log~0(~m/cm3mol-1)=E+

(6)

are listed in table 1. Relative volume reduction factors V&/V& evaluated at (*pZJ” = 2.0 are given in table 2.

5. Entropy reduction

factors

The entropy reduction

factor S*(n) is related to

the other reduction factors by: S*(n)=N*c(“)k=p’(n)

P(n)lP(n).

(7)

3~6) is the number of external degrees of freedom possessed by a molecule of chain length ?I, and NA and k are the Avogadro and Boltzmann constants. Fig. 3 shows plots of cS,Jc,+a and c,&cA,~ against the number of repeating units in the chain (viz. the number of silicon atoms in the dimethylsiloxane, or

Fig. 3. The chain length dependence of the parameter c(n). The quantity c,/cA,~ is plotted against the number of silicon atoms in the dimethylsiloxaae (curve 1) or carbon atoms in the n-zdkane (curve 2).

the number of carbon atoms in the nmalkane). Many of the characteristic bulk physical properties of the polydimetbylsiloxanes [2] have been attributed to their greater chain flexibility, a view which is consistent with their high c values relative to the n-alkanes. There is, however, a great deal of arbitrariness in the choice of the scale on the abscissa in fig. 3 t, and, while it is true that a dimethylsiloxane oligomer has considerably more external degrees of freedom per repeating unit than a n-alkane, it is nevertheless impossible to divorce those contriiutions toS* due to side-group rotation from those which arise from a greater flexibility in the backbone of the chain.

t Another possibility would be to use the number of backbone atoms as the variable representing the siloxane chain

len,$h. Tl~iswould have the effect of reducing by a factor of 2 the value of &g,@t; significantiy, howwer. it would Still even thea be much kget than do?,&&.

21

E. Dickinson et ~L~S~JWZ~ ~~rmponding states n&ions 6. Temperature dependence of the isobaric expansivity

A very stringent test of the principle of corresponding states is provided by plotting apt cr.,,T the dimensionless reduced quantity 5 (daJd7). Superimposable curves should be obtained for ah pure liq uids which obey the principle. As is shown in fig. 4, the dimethyl siloxanes comply with this criterion to within only about 10 per cent in (d -‘/dn. The average siloxane curve, labelled 1 in fig. 7 , is obtained by differentiating eq. (2) with respect to temperature and then using a mean B value of 8.91; it is displaced distinctly away from curve 2, the analogous average line for the normal alkanes [7]. It is Seen from this test that the dimethylsiloxanes follow a corresponding states principle which is slightly different from that adhered to by the n-alkanes. Clearly, part of the discrepancy between curves 1 and 2 of fig. 4 is attributable to experimental artefacts: the test involves double differentiation of smoothed specific volume data, with the result that the exact form of the curves - especially at the ends - is somewhat artificial and depends upon particularities of the data fitting procedure. (The inclusion of an extra

0.3

04

polynomial coefficient in the fitting expression for the molar volume changes the shapes of the curves a little [ 131, but does not sensibly alter the fii broad conclusions; the use of only three coefficients, however, requiring dap/dT to be temperature independent, leads to absurd results.) The above evidence would seem to suggest that the n&canes and dtmethylsiloxanes follow dissimilar reduced equations of state. None&less, we believe that this difference is insufftcient to preclude meaningful analysis of mixtures of n-alkaneswith dimethyl-

siloxanes using approacheswhich assume, however implicitly. a strict adherence to a unique principle of corresponding states. On the other hand. small discrepancies between experimental thermodynamic data and the predictions from such approaches should be considered in the light of those deviations from a universal principle which are presented here. As well as being a sensitive test of the observed difference in thermodynamic behaviour between the two sets of oligomers, the reduced quantity s2(d%ldT) may also be used to check the reasonableness of theoretical equations of state derived from statistical thermodynamic models of an assembly of chain molecules. Patterson and Bardin have compared [7] the

0.5

0.7

I 0.7

(AT

Plots of o;z(&p/dT) againstupT; (a) dimer, (b) tier. siloxane,(2) average n-alkane. Fig. 4.

(e) temu.

(d) pentamer. (e) hexamer, (1) averagedimethyl-

:

22.”

,.

.’

.

_

__

.

E Dkktkwn et aLbdoxa~~ wrmpanfiing nates rdations

”

&er.& n-alkaile behaviour (curve 2 of fig. 4) with given by ‘several tid-uced equations of state, in-cliiw those of Orwell and Flory [I l] [eq. (S)], Sia kd Somcynsky [14]‘and van der Waals. The van’& Weals equation was found to overestimate $‘i%/dT) by a factor of 7 at low values of QpT, reducing to a factor of 2 at the higher values. The FloG equation was in much better agreement at high values of %T (within 30 per cent), but was almost as bad as the van der Waals equation at the lower values. Most successful was the hole theory of Sii and Somcynsky; however, while it gave correct values at ST” 0.4, it led to large deviations at the extremes. As can be seen in fig. 4, our dimethylsiloxane experimental equation of state differs from the theoretical equations in a manner qualitatively similar to that of the n-alkanes. (In fact, the extent of the disagreement tends to be greater for the siloxanes, whose mean B value is smaller than its n-alkane counterpart.) Clearly, the experimental equations of state of the two sets of oligomers are in much greater conformity

References

with each other than they are with any of the theoretical equations, a conclusion which must cast doubt

and P. Picker, Trans. Faraday Sot. 64 (1968) 648. [ 151 AJ. Pretty, Ph.D. Thesis, The University, Sheffield (1973). [14] R. Simha and T. Somcynsky, Macromol. 2 (1969) 342.

I’ tl$

upon the applicability of the models of Flory and Siia to the low reduced temperature region of an oligomeric liquid.

Atiowkdgement

E.D., ALP. and P.A.S. acknowledge the receipt of S.R.C. Studentships.

[I] R. Simha and A.J. Havlik, J. Amer. Chem. Sot 86 (1964) 197; H. Shihand P.J. Rory. Maaomot 5 (1972) 758. [2] E Dickiu~~n and LA. ?&Lure. JCS. Faraday I70 (1974) 2328. 131E. Didcinson, LA. McLure and B.H. Powell. J.C.S. Faraday 170 (1974) 2321. [4] J.S. Rowlinson. Liquids and Liquid Mixtures. 2nd Ed. (Butterworth. London, 1969) chap. 9. [51 T.W. &land, J.S. Rowlinson and GA. Sather, Trans. Faraday Sot. 64 (1968) 1447. [6] I. Prigoginc, Molecular Theory of Solutions (NorthHolland, Amsterdam, 1957), chap. 16 and 17. 171 D. Patterson and J.M. Bardin. Trans. Faraday Sot. 66 (1970) 321. IS] LA. McLure and AJ. Pretty. J. Chem. Eng. Data, to be published. 19) LA McLwe. AJ. Pretty and P.A. Sadler, J. Chem. Eng. Data, to be published. [101E. Dickinson, PILD. Thesis, The University, Sheffield

(1972). chap. 9. [ 111 R.A. Orwell and PJ. Flory, J. Amer. Chem. Sot. 89 (1967) 6814.

[ 12) D. Patterson, S.N. Bhattachasyya